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You are the treasurer of the new island kingdom Polar Koordinatea where no calculators are allowed....

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Advance Math

You are the treasurer of the new island kingdom PolarKoordinatea where no calculators are allowed. The Queen summons youand gives you 24h to design a coin for the kingdom. After asleepless night, you come up with two proposals: (i) The coin isthe circle r = 1, having inside the rose r = cos(3?) whose petalsare plated in gold; (ii) The coin is circle r = 1, having insidethe rose r = sin(5?) whose petals are plated in gold. (a) Draw thetwo designs you submit to the Queen. (b) Seeing the designs, theQueen decides: “Make the one that has more gold. Bring ittomorrow!” What do you tell the Queen the next day? (c) Next day,after your answer, the Queen decides again: “You modify the design(i) as follows. Inside the coin r = 1, draw also the circle r =1/2. The part of the rose r = cos(3?) which is inside r = 1/2 shallbe covered with platinum, the rest of the rose shall be coveredwith gold. And this shall be the coin of Polar Koordinatea!” Justwhen you are about to leave happy, the Queen says: “I would like toknow by tomorrow if more platinum or more gold is needed for thenew coin. Tell me please the exact difference between the twoareas.” Can you answer this?

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